On multipole approximations of the Fokker-Planck-Landau operator

  • Mohammed Lemou
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


In this paper we are concerned with numerical approximations of the Fokker-Planck-Landau operator which is used to describe collisions between charged particles in a plasma. Our aim is to construct accurate approximations to this operator that have a reduced numerical complexity and still satisfy some important physical properties of conservation and entropy. After a brief description of some recent works on the discretizations of such an operator, we focus on the application of the well-known Fast Multipole Method (FMM) to the approximation of the three dimensional Fokker-Planck-Landau operators and review the results in [18]. In the same spirit but in the simpler case of a spherical geometry, we give a short presentation of an alternative method that uses wavelet approximation techniques. Then, we check the efficiency of the multipole method in terms of accuracy and computational cost and present some numerical tests at the end of the paper.


Collision Operator Entropy Inequality Multipole Expansion Fast Multipole Method Wavelet Approximation 
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  1. 1.
    Alpert, B., Beylkin, G., Gines, D., and Vozovoi, L.: Adaptive solution of partial differential equations in multiwavelet bases. J. Comput. Phys., 182 (1), 149–190, (2002).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Antoine, X., and Lemou, M.: Multiwavelet approximations of a collision operator in kinetic theory. Submitted.Google Scholar
  3. 3.
    Arsene’v, A. A., and Buryac, O. E.: On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation. Math. USSR Sbornik 69 (2), 465–478, (1991).CrossRefGoogle Scholar
  4. 4.
    Beylkin, G., Coifman, C, and Rokhlin, V.: Fast wavelet transforms and numerical algo-rithms. I. Comm. Pure Appl. Math., 44 (2), 141–183, (1991).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bokanowski, O., and Lemou, M: Fast Multipole Method for multidimensional integrals. CR. Acad. Sci. Paris, Série I, 326, 105–110, (1998).Google Scholar
  6. 6.
    Bokanowski, O., and Lemou, M.: Fast Multipole Method for multi-variable integrals. To appear in SIAM, J. Num. Anal.Google Scholar
  7. 7.
    Buet, C, and Cordier, S.: Conservative and entropy decaying numerical scheme for the isotropic Fokker-Planck-Landau operator. J. Comput. Phys., 145 (1), 228–245, (1998).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Buet, C., Cordier, S., Degond, P., and Lemou, M.: Fast algorithms for numerical, conservative and entropy approximations of the Fokker-Planck-Landau equation, J. Comput. Phys., 133, 310–322, (1997).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chacón, L., Barnes, D. C, Knoll, D. A., and Miley, G. H.: An implicit energy-conservative 2D Fokker-Planck algorithm. I. Difference scheme. J. Comput. Phys., 157 (2), 618–653, (2000).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chacón, L., Barnes, D. C., Knoll, D. A., and Miley, G. H.: An implicit energy-conservative 2D Fokker-Planck algorithm. II. Jacobian-free Newton-Krylov solver. J. Comput. Phys. 157 (2), 654–682, (2000).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Degond, P., andLucquin-Desreux, B.: An entropy scheme for the Fokker-Planck collision of plasma kinetic theory. Numer. Math. 68, 239–262, (1994).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Degond, P., and Lucquin-Desreux, B.: The Fokker-Planck asymptotics of the Boltzmann collision operator in the coulomb case. Math. Models and Methods in Appl. Sci, 2 (2), 167–182,(1992).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Desvillettes, L.: On asymptotics of the Boltzmann equation the collisions become grazing. Trans JH. and Stat. Phys., 21 (3), 259–276, (1992).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Epperlein, E. M: Implicit and conservative difference scheme for the Fokker-Planck equation, J. Comput. Phys. , 112, 291–297, (1994).MATHCrossRefGoogle Scholar
  15. 15.
    Filbet, F, and Pareschi, L.: A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the non homogeneous case. To appear in J. Comput. Phys. (Available online at
  16. 16.
    Greengard, L., and Rokhlin, V.: A fast algorithm for a particle simulation. J. Comput. Phys. 73, (1987).Google Scholar
  17. 17.
    Greengard, L., Rokhlin, V.: The rapid evaluation of potential fields in three dimensions. Vortex Methods, Anderson, C. , Greengard, C. (Eds.) Lecture Notes in Mathematics, Springer-Verlag, N.Y., (1988).Google Scholar
  18. 18.
    Lemou, M.: Multipole expansions for the Fokker-Planck-Landau operator. Numer. Math. 78, 597–618, (1998).MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Lemou, M.: Numerical algorithms for axisymmetric Fokker-Planck-Landau operators. J. Comput. Phys., 157 (2), 762–786, (2000).MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Lemou, M.: Exact solutions of the Fokker-Planck equation. CR. Acad. Sci. 319, Serie 1,579–583,(1994).Google Scholar
  21. 21.
    Pareschi, L., and Perthame, B.: A Fourier spectral method for homogeneous Boltzmann equations. Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994). Transport Theory Statist. Phys., 25 (3–5), 369–382, (1996).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Pareschi, L., Russo, G., and Toscani, G.: Fast Spectral Methods for the Fokker-Planck-Landau Collision Operator. J. Comput. Phys., 165 (1), 216–236, (2000).MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Mohammed Lemou
    • 1
  1. 1.MIP, UMR 5640CNRS et Université Paul SabatierToulouseFrance

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